Which of the following sets of numbers has the greatest standard deviation?
Show Hint
- 2, 3, 4
- 2.5, 3, 3.5
- 1, 1.25, 1.5
- -2, 0, 2
- 20, 21, 21.5
The Correct Option is D
Solution and Explanation
Standard deviation is a measure of the amount of variation or dispersion of a set of values. A low standard deviation indicates that the values tend to be close to the mean (also called the expected value) of the set, while a high standard deviation indicates that the values are spread out over a wider range. We do not need to calculate the exact standard deviation; we can compare the "spread" of each set.
Step 2: Detailed Explanation:
Let's analyze the spread of each set of numbers. A simple way to estimate spread is to look at the range (maximum value minus minimum value) and how the numbers are distributed around their mean.
(A) 2, 3, 4: The mean is 3. The numbers are 1 unit away from the mean. The range is \(4 - 2 = 2\).
(B) 2.5, 3, 3.5: The mean is 3. The numbers are 0.5 units away from the mean. The range is \(3.5 - 2.5 = 1\). This set is less spread out than (A).
(C) 1, 1.25, 1.5: The mean is 1.25. The numbers are 0.25 units away from the mean. The range is \(1.5 - 1 = 0.5\). This set is very tightly clustered.
(D) -2, 0, 2: The mean is 0. The numbers are 2 units away from the mean. The range is \(2 - (-2) = 4\). This set is quite spread out.
(E) 20, 21, 21.5: The mean is 20.833... The absolute values of the numbers are large, but they are close to each other. The range is \(21.5 - 20 = 1.5\). The spread is smaller than in set (D).
Step 3: Comparing the Spreads:
By comparing the ranges and the distances of the points from their respective means, we can see that set (D) has the values that are furthest from their center.
- Spread of (A): distances from mean are \{1, 0, 1\}
- Spread of (B): distances from mean are \{0.5, 0, 0.5\}
- Spread of (C): distances from mean are \{0.25, 0, 0.25\}
- Spread of (D): distances from mean are \{2, 0, 2\}
- Spread of (E): distances from mean are roughly \{0.83, 0.17, 0.67\}
The deviations from the mean are largest for set (D). Therefore, it has the greatest standard deviation.
Step 4: Final Answer:
The set \{-2, 0, 2\} has the greatest spread and thus the greatest standard deviation.
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